In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a diagonal is a
line segment joining two
vertices of a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
or
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
, when those vertices are not on the same
edge
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed ...
. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the
ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...
διαγώνιος ''diagonios'', "from angle to angle" (from διά- ''dia-'', "through", "across" and γωνία ''gonia'', "angle", related to ''gony'' "knee"); it was used by both
Strabo and
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
to refer to a line connecting two vertices of a
rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
or
cuboid, and later adopted into Latin as ''diagonus'' ("slanting line").
In
matrix algebra
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
, the diagonal of a square
matrix consists of the entries on the line from the top left corner to the bottom right corner.
There are also other, non-mathematical uses.
Non-mathematical uses
![2512-échafaudage-Réunion](https://upload.wikimedia.org/wikipedia/commons/0/0e/2512-%C3%A9chafaudage-R%C3%A9union.jpg)
In
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, a diagonal brace is a beam used to brace a rectangular structure (such as
scaffolding
Scaffolding, also called scaffold or staging, is a temporary structure used to support a work crew and materials to aid in the construction, maintenance and repair of buildings, bridges and all other man-made structures. Scaffolds are widely use ...
) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations diagonal braces are often not connected to the corners of the rectangle.
Diagonal pliers
Diagonal pliers (also known as wire cutters, diagonal cutting pliers, diagonal cutters, side cutters, dikes or Nippy cutters) are pliers intended for the cutting of wire (they are generally not used to grab or turn anything). The plane defi ...
are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or "on a diagonal", hence the name.
A
diagonal lashing
A lashing is an arrangement of rope, wire, or webbing with linking device used to secure and fasten two or more items together in a somewhat rigid manner. Lashings are most commonly applied to timber poles, and are commonly associated with cargo, ...
is a type of lashing used to bind spars or poles together applied so that the lashings cross over the poles at an angle.
In
association football
Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...
, the
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch.
Polygons
As applied to a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
, a diagonal is a
line segment joining any two non-consecutive vertices. Therefore, a
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
has two diagonals, joining opposite pairs of vertices. For any
convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
, all the diagonals are inside the polygon, but for
re-entrant polygon
A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive.
Polyg ...
s, some diagonals are outside of the polygon.
Any ''n''-sided polygon (''n'' ≥ 3),
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
or
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity of a ...
, has
diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or ''n'' − 3 diagonals, and each diagonal is shared by two vertices.
Regions formed by diagonals
In a
convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
, if no three diagonals are
concurrent
Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to:
Law
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion (also called a "concurrence"), a ...
at a single point in the interior, the number of regions that the diagonals divide the interior into is given by
:
For ''n''-gons with ''n''=3, 4, ... the number of regions is
:1, 4, 11, 25, 50, 91, 154, 246...
This is
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
sequence A006522.
Intersections of diagonals
If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by
.
beginning at 2:10 This holds, for example, for any
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
with an odd number of sides. The formula follows from the fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of the ''n'' vertices four at a time.
Regular polygons
There exist separate formulae for calculating the length of the longest diagonal in regular polygons with an even or odd number of sides.
In an even-sided regular polygon with ''n'' sides and side length ''a'', the length of the longest diagonal is equivalent to the diameter of its circumcircle because the long diagonals all intersect each other at the polygon's center. This is given by the following formula.
:
The length of the longest diagonal of any odd-sided regular ''n''-sided polygon (''n'' ≥ 5) with side length ''a'' is given by the following formula.
:
The length of a polygon's shortest diagonal can also be calculated for all polygons (''n'' ≥ 4) with the following formula. As the number of sides approaches infinity, the shortest diagonal approaches ''2a''.
:
These are not applicable for a
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
which has no diagonals.
Special cases include:
A
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is
A
regular pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
has five diagonals all of the same length. The ratio of a diagonal to a side is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
,
A regular
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
has nine diagonals: the six shorter ones are equal to each other in length; the three longer ones are equal to each other in length and intersect each other at the center of the hexagon. The ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is
.
A regular
heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.
The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived nu ...
has 14 diagonals. The seven shorter ones equal each other, and the seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a short and a long diagonal.
In general a regular n-gon has
distinct diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.
Polyhedrons
A
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
(a
solid object in
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
, bounded by
two-dimensional
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
) may have two different types of diagonals:
face diagonals on the various faces, connecting non-adjacent vertices on the same face; and
space diagonal
In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with '' face diagonals'', which connect vertices on the same face (bu ...
s, entirely in the interior of the polyhedron (except for the endpoints on the vertices).
Just as a
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
has no diagonals, so also a
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
(with four triangular faces) has no face diagonals and no space diagonals.
A
cuboid has two diagonals on each of the six faces and four space diagonals.
Matrices
For a
square matrix, the ''diagonal'' (or ''main diagonal'' or ''principal diagonal'') is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix
with row index specified by
and column index specified by
, these would be entries
with
. For example, the
identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:
:
The top-right to bottom-left diagonal is sometimes described as the ''minor'' diagonal or ''antidiagonal''.
The ''off-diagonal'' entries are those not on the main diagonal. A ''
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
'' is one whose off-diagonal entries are all zero.
A ''superdiagonal'' entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those
with
, the superdiagonal entries are those with
. For example, the non-zero entries of the following matrix all lie in the superdiagonal:
:
Likewise, a ''subdiagonal'' entry is one that is directly below and to the left of the main diagonal, that is, an entry
with
.
General matrix diagonals can be specified by an index
measured relative to the main diagonal: the main diagonal has
; the superdiagonal has
; the subdiagonal has
; and in general, the
-diagonal consists of the entries
with
.
Geometry
By analogy, the
subset of the
Cartesian product ''X''×''X'' of any set ''X'' with itself, consisting of all pairs (x,x), is called the diagonal, and is the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of the
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
relation on ''X'' or equivalently the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
from ''X'' to ''X''. This plays an important part in geometry; for example, the
fixed points of a
mapping ''F'' from ''X'' to itself may be obtained by intersecting the graph of ''F'' with the diagonal.
In geometric studies, the idea of intersecting the diagonal ''with itself'' is common, not directly, but by perturbing it within an
equivalence class. This is related at a deep level with the
Euler characteristic and the zeros of
vector fields. For example, the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
''S''
1 has
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...
s 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
''S''
1xS
1 and observe that it can move ''off itself'' by the small motion (θ, θ) to (θ, θ + ε). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the
Lefschetz fixed-point theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named ...
; the self-intersection of the diagonal is the special case of the identity function.
See also
*
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
*
Main diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matri ...
*
Diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct al ...
Notes
References
*
*
*
*
External links
{{Wiktionary, diagonal
Diagonals of a polygonwith interactive animation
from
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...
.
Diagonalof a matrix from
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...
.
Elementary geometry